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INDIANA UNIVERSITY 



conditions that must be satisfied hj the density in order that Pois- 

 son's equation should hold. The proofs given by Diriehlet^ Poin- 

 care^ and Wangerin^^, assume that the density has finite first de- 

 rivatives. Dirichlet really proves that Poisson 's equation is true at 

 the center of a sphere of variable density if the density satisfies 

 the above condition. By surrounding any point of a given distri- 

 bution ^vith a sphere and making use of Laplace's equation for the 

 matter outside this, Poisson 's equation is seen to be true at all 

 points. Later proofs by Llolder'^, iMorera^-, and Petrini^^, have 

 lessened the conditions that must be imposed upon the density. 



11. 



Proof by Gauss's theorem of the arithmetic mean. This theo- 

 rem, which is made the basis of the following proof of Poisson 's 

 equation, states that: The average value of the potential function 

 on the surface of a sphere is the same as the value aA the center of 

 the sphere if the matter is without the sphere, and if the matter is 

 ivithin the spin )■( if has the same value that it ^roiiM have if all of 

 that matter were conct titrated at its coite)-. 



This theorem can be established without the use of Gauss's the- 

 orem of the integral of the normal component of the force over a 

 closed surface by direct integration^^ 



To apply the theorem, take the point in ([uestion as origin. 0, 

 and develop the potential by Taylor's theorem. We have then that 

 in the vicinity of 0. 



do: ^ dx dxdlt 

 where the subscript 0 means evaluation at the origin. 



About 0 construct a small sphere of radius E and integrate the 

 value of — T^„ over its surface, giving 



ffiV - V^dS - li^JjxdS + ~y:(^.)Jfx'^dS -f i-(|^)o//'a-//f^>S 



ox ' ox 0X01/ 



+ ' U) 



s p. Lejeune-Dii-iehlet, Yoi Je-saui/cii Hhcr die Lit amgcl-eltrlen rcrhalfiii.'a <l. Quad- 

 rats d. Enlfenmng wiricendeii KrUfu:, hrsfj. v. F. Grulje, 1887. SO. 

 9 H. Poincaro, TJieorie du Potenticl ycftonien, p. 8S. 



A. Wangerin, Tl'eorle dcs Potenliuls mid der Kurjidfi'uktioiieii I, p. 73. 

 1^0. Holder, Diss. Tiibiiifjoi, lS8ii. p. lo. 

 12 G. Morera, Lomh. 1st Bend. (2) Vol, 20, p. 302. 



" H. Petrini. K. Vet. Al:ad. Oefrers. Stockholm, 1899; Act,i M<i Hiematica, Vol. 

 31, 1908, p. 127. 



14 peirce. Neict. Pot. Panel., p. Hs. I'circe shows it by direct calculation only for 

 the case of exterior matter. A mere change in limits will make his same integral 

 apply to interior matter. 



